a level maths - coordinate geometry
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This book is under copyright to A-level Maths Tutor. However, it may bedistributed freely provided it is not sold for profit.
Contents
the line between two points 3
more about straight lines 9
parametric equations 17
circles & ellipses 20
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The Line Between Two Points
Distance of a line between two points
Triangle ABC is a right angled triangle.
therefore
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Example
What is the distance between points A(5,6) and B(-4,-3) correct to 2 d.p.?
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Finding the mid-point of a line between two points
The x-y coordinates of the midpoint M between two point A, B is found by taking the
average of the x-coordinates(x1, x
2), then repeating for the y-coordinates(y
1, y
2)
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Example
What are the coordinates of the mid-point of the line joining the coordinates (4,7) and
(-8,8)?
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Finding the gradient of a line
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Example
To 2 decimal places, what is the gradient of the line joining the coordinates (-5,6) and
(9,-7)?
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More About Straight Lines
Parallel lines
Parallel lines make equal corresponding angles() with the x-axis.
Therefore their gradients are equal.
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Perpendicular lines
If two lines are perpendicular to eachother, the product of their gradients is -1.
If the gradient of AB is m1and the gradient of CD is m2, then:
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Equation of a straight line y = mx + c
The equation of a straight line is given by:
mis the gradient of the line
cis the intercept on the y-axis
Example
What is the equation of the straight line with gradient 3 that crosses the y-axis at y= -3 ?
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Finding the intersection point between two straight lines
There are two types of problem here. One where the lines are not perpendicular to
eachother and the other when they are.
To solve the former all that is needed is to solve the equations of the lines simultaneously.
With the later, only one equation is given and the second equation must be worked out
from the information supplied. then it is a matter of proceding as before ie to solve the two
equations simultaneously.
Example #1
Find the intersection point of the two straight lines:
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Example #2
A straight liney = 2x + 4.5 intersects another perpendicularly. If the second straight line
has an intercept of -0.5 on the y-axis, what are the coodinates of the point of intersection
of the two lines? (answer to 1 d.p.)
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Finding the eq. of a straight line from one point + gradient
Solution is by using the expression for gradient(m) for an actual point(x ,y1 1) and a
generalized point(x,y).
The straight line equation is found by substituting values ofx , y1 1and minto the above.
Example
A line of gradient 3 passes through a point (2,5). What is the equation of the line?
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Finding the equation of a straight line from two points
Solution is by first finding the gradient mfrom the x and y values from the points (x ,y1 1)
and (x ,y2 2)
Then we use the expression again, but this time with one actual point and a generalized
point(x,y).
The straight line equation is found by substituting forx , y1 1and m.
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Example
Find the equation of the line between the two points (2,3) and (-5,7).
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Parametric Equations
Introduction
There is another way of writing y as a function of x and that is to use two separate
equations.
One equation has x as a function of t (or ) eg x = 2t
and the other equation has y as function of t (or ) eg y = t2
The variables t, are called parametersand the two equalities parametric equations.
Common questions on this topic are the plotting of parametric equations and their
conversion to a single Cartesian equation.
Example #1
Plot the graph of the curve given parametrically by the equations:
x = 2cos y = 2sin
o 0 45 90 135 180
x = 2cos 2 1.414 0 -1.414 -2
y = 2sin 0 1.414 2 1.414 0
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Example #2
What is the Cartesian equation given parametrically by:
x = t2+3 y = t3+ 3t
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Example #3
What is the Cartesian equation given parametrically by:
x = 2sin y = 2sin2
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Circles & Ellipses
Circles
Any point P is described by Pythagoras' Theorem. So the equation of a circle with centre
(0,0)and radius ris given by:
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or in terms of parameters,
For a circle with its centre off-set from the origin at a point C(a,b), again, by Pythagoras,
the equation is given by:
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Circle equation expanded(usual form)
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Example
What is the radius and the coodinates of the centre of the circle with equation:
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Ellipses
The maximum displacement (a) along the x-axis is called the semi-major axis, while the
maximum displacement along the y-axis (b) is called the semi-minoraxis. This is when
a>b. When b>a the names are interchanged.
The eccentricity(e) of an ellipse is defined as:
From the eccentricity we can define the points of focus (plural foci):
F1(ae,0) and F2(-ae,0)
and the directrices(directrix lines) at x=a/e and x=-a/e.
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The directrices are two special lines parallel to the y-axis and either side of it(when the
ellipse is centred at the origin).
Their unique property concerns the ratio of the distance between a point(P1)on the curve
to a focal point(F1) and a line from the point to the directrix.
The ratio gives the eccentricity 'e' .
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Notes
This book is under copyright to A-level Maths Tutor. However, it may be
distributed freely provided it is not sold for profit.
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